Let X and X' denote a single set in the two topologies T and T', respectively. Let i:X'-> X be the identity function(adsbygoogle = window.adsbygoogle || []).push({});

a. Show that i is continuous <=> T' is finer than T.

b. Show that i is a homeomorphism <=> T'=T

This is all i've got.

According to the first statement... X [tex]\subset[/tex] T and X' [tex]\subset[/tex] T

a. if i is continuous... each open subset V of X' the set i^-1 is an open subset of X

T' is finer than T means... T [tex]\subset[/tex] T'.

i don't know where to go from here...

b. if i is a homeomorphism...

then... i is... a bijection therefore the function and the inverse function are continuous.

i and i^-1 are continuous. each open subset V of X' the set i^-1 is an open subset of X

if T' = T... then.....

i don't know where to go from here either...

can someone help me out?

Thank You,

tomboi03

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# I: X'->X the identity function with topology

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